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Chapter 5

• Learning Objectives• Principles Used in this Chapter

1. Using Time Lines2. Compounding and Future Value3 Di ti d P t V l3. Discounting and Present Value4. Making Interest Rates Comparable

1. Construct cash flow timelines to organize your analysis of time value of money problems and learn three techniques for solving time value of money problems.U d t d di d l l t th2. Understand compounding and calculate the future value of cash flow using mathematical formulas, a financial calculator and an Excel worksheet.

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3. Understand discounting and calculate the present value of cash flows using mathematical formulas, a financial calculator and an excel spreadsheet.

4. Understand how interest rates are quoted and how to make them comparable.

Principle 1: Money Has a Time Value.

The concept of time value of money – a dollar received today other things being the same is worth more thantoday, other things being the same, is worth more than a dollar received a year from now, underlies many financial decisions faced in business.

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A timeline identifies the timing and amount of a stream of cash flows along with the interest rate.

A i li i i ll d i b i◦ A timeline is typically expressed in years, but it could also be expressed as months, days or any other unit of time.

i=10%

YearsCash flow -$100 $30 $20 -$10 $50

0 1 2 3 4

The 4-year timeline illustrates the following:◦ The interest rate is 10%.◦ Cash outflow of $100 occurs at the beginning of the first year (at time 0), ◦ Cash inflows of $30 and $20 in years 1 and 2, ◦ Cash outflow of $10 in year 3◦ Cash inflow of $50 in year 4.

Creating a TimelineSuppose you lend a friend $10,000 today to help him finance a new Jimmy John’s Sub Shop franchiseIn return he promises to give you $12,155 at the end of the fourth year. How can one represent this as a timeline?

Note that the interest rate is 5%.

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Draw a timeline for an investment of $40,000 today that returns nothing in one year, $20,000 at the end of year 2, nothing in year 3, and $40,000 at the end of year 4.

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i=interest rate; not givenTime Period

Cash flow -$40,000 $0 $20,000 $0 $40,000

0 1 2 3 4

What is the difference between simple interest and compound interest?◦ Simple interest: Interest is earned only on the

principal amount.◦ Compound interest: Interest is earned on both the◦ Compound interest: Interest is earned on both the

principal and accumulated interest of prior periods.

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Suppose that you deposited $500 in your savings account that earns 5% annual interest. How much will you have in your account after two years using

(a) simple interest and◦ (a) simple interest and ◦ (b) compound interest?

Simple Interest◦ Interest earned = 5% of $500 = .05×500 = $25 per year◦ Total interest earned = $25×2 = $50◦ Balance in your savings account:◦ Balance in your savings account: = Principal + accumulated interest = $500 + $50 = $550

Compound interest◦ Interest earned in Year 1 = 5% of $500 = $25◦ Interest earned in Year 2

5% of ($500 + accumulated interest) = 5% of ($500 + accumulated interest) = 5% of ($500 + 25) = .05×525 = $26.25◦ Balance in your savings account: = Principal + interest earned = $500 + $25 + $26.25 = $551.25

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Time value of money calculations involve Present value ◦ (what a cash flow would be worth to you today) and

Future value ( h t h fl ill b th i th f t )◦ (what a cash flow will be worth in the future).

In example 5.1, Present value is $500 and Future value is $551.25.

We can apply equation 5-1a to example 5.1

FV2 = PV(1+i)n

= 500(1.05)2= $551.25

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Continue example 5.1 where you deposited $500 in savings account earning 5% annual interest.

Show the amount of interest earned for the fi t fi d th l f ifirst five years and the value of your savings at the end of five years.

YEAR PV or Beginning Value

Interest Earned (5%) FV or Ending Value

1 $500.00 $500*.05 = $25 $525

2 $525.00 $525*.05 = $26.25 $551.25

3 $551 25 $551 25* 05 =$27 56 $578 813 $551.25 $551.25*.05 =$27.56 $578.81

4 $578.81 $578.81*.05=$28.94 $607.75

5 $607.75 $607.75*.05=$30.39 $638.14

We get the same answer from the FV equation FV = PV(1+i)n

◦ = 500(1.05)5◦ = $638.14

So the balance in savings account at the end of 5 years will equal $638.14.

The total interest earned on the original principal amount of $500 will equal $138.14 ◦ ($638.14 -$500.00).

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Figure 5.1 Future Value and Compound Interest Illustrated

Figure 5.1 Future Value and Compound Interest Illustrated

1. Power of Time: Future value of original investment increases with time.

However, there will be no change in future value if the interest rate is equal to zero.

2. Power of Rate of Interest: An increase in rate of interest leads to an increase in future value.

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Calculating the Future Value of a Cash FlowYou are put in charge of managing your firm’s working capital. Your firm has $100,000 in extra cash on hand and decides to put it in a savings account paying 7% interest compounded annually How much will youinterest compounded annually. How much will you have in your account in 10 years?

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What is the future value of $10,000 compounded at 12% annually for 20 years?

i=12%YearsCash flow -$10,000

0 1 2 … 20

Future Value=?

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This is a simple future value problem.

We can use the FV Equation.

Solve using the Mathematical Formula

FV = $10,000(1.12)20

= $10,000(9.6463)= $96,462.93

Solve using Financial CalculatorEnter:

N = 20 I/Y = 12% PV is negative as it is a

cash outflowPV = -10,000 PMT = 0FV = $96,462.93

cash outflow

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If you invest $10,000 at 12%, it will be worth $96,462.93 in 20 years.

Example 5.2 A DVD rental firm is currently renting 8,000 DVDs per year.

How many DVDs will the firm be renting in 10 years if the demand for DVD rentals is

t d t i b 7% ?expected to increase by 7% per year?

Using FV Equation ◦ FV = 8000(1.07)10 = $15,737.21

Example 5.3 Your annual tuition at a State University is currently $20,000.

If the tuition increases by 6% annually, what will be the annual cost of attending the State U i it i 25 ?University in 25 years?

FV =$20,000 (1.06)25= $85,837.41 per year

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Interest rates are quoted as Annual Percentage Rates (APR)

Banks frequently offer savings account that compound interest every day, month, or

tquarter. Credit cards compound interest monthly◦ Some compound daily

More frequent compounding will generate a higher interest rate if the APR is the same.

Example 5.4 You invest $500 for seven years to earn an annual interest rate of 8%, and the investment is compounded semi-annually. What will be the future value of this investment?investment?

We will use equation 5-1b to solve the problem.

This equation adjusts the number of compounding periods and interest rate to reflect the semi-annual compounding.

FV = PV(1+i/2)m*2

= 500(1+.08/2)7*2

= 500(1.7317)= $865.85

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Calculating Future Values Using Non-Annual Compounding PeriodsYou have been put in charge of managing your firm’s cash position and noticed that the Plaza National Bank of Portland, Oregon, has recently decided to begin paying interest compounded semi-annually instead of annually. If you deposit $1,000 with Plaza National Bank at an interest rate of 12%, what will your account balance be in five years?

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If you deposit $50,000 in an account that pays an annual interest rate of 10% compounded monthly, what will your account balance be in 10 years?

i=10%Months

Cash flow -$50,000

0 1 2 … 120

FV of $50,000Compounded for 120 months @ 10%/12

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This involves solving for future value of $50,000

The only difference is that the interest is d d hlcompounded monthly.

We will use equation 5-1b.

Using Mathematical Formula

FV = PV (1+i/12)m*12

= $50,000 (1+0.10/12)10*12

= $50,000 (2.7070)= $135,352.07

Using a Financial CalculatorEnter:

N = 120 I/Y = .833% PV 50 000PV = -50,000 PMT = 0FV = $135,298.39

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More frequent compounding leads to a higher FV as you are earning interest more often on interest you have previously earned.

f h d d ll h If the interest was compounded annually, the FV would have been equal to: ◦ $50,000 (1.10)10 = $129,687.12

What is value today of cash flow to be received in the future?

◦ The answer to this question requires computing the present value i e the value today of a future cash flowpresent value i.e. the value today of a future cash flow, and the process of discounting, determining the present value of an expected future cash flow.

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The term in the bracket is known as the Present Value Interest Factor (PVIF).

PV = FVn PVIF

The PV equation and the FV equation are the same

FV = PV(1 + i)nPV FV/(1 + i)n PV = FV/(1 + i)n

Example 5.5 How much will $5,000 to be received in 10 years be worth today if the interest rate is 7%?

( ( ) ) PV = FV (1/(1+i)n )= 5000 (1/(1.07)10)= 5000 (.5083)= $2,541.50

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If the interest rate (discount rate) is higher (say 9%), the PV will be lower.◦ PV = 5000*(1/(1.09)10) = 5000*(0.4224)

=$2,112.00 If the interest rate (discount rate) is lower If the interest rate (discount rate) is lower

(say 2%), the PV will be higher.◦ PV = 5000*(1/(1.02)10) = 5000*(0.8203)

= $4,101.50

Solving for the Present Value of a Future Cash FlowYour firm has just sold a piece of property for $500,000, but under the sales agreement, it won’t receive the $500,000 until ten years from today. What is the present value of $500,000 to be received ten years from today if the discount rate is 6% annually?today if the discount rate is 6% annually?

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What is the present value of $100,000 to be received at the end of 25 years given a 5% discount rate?

i=5%Years

Cash flow $100,000

0 1 2 … 25

PresentPresent Value =?

Here we are solving for the present value (PV) of $100,000 to be received at the end of 25 years using a 5% interest rate.

l We can solve using equation 5-2.

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Using the Mathematical Formula

PV = $100,000 [1/(1.05)25)= $100,000 [0.2953]= $29,530

Using a Financial CalculatorEnter:

N = 25 I/Y = 5 PMT 0PMT = 0FV = 100,000PV = -$29,530

Key Question: How long will it take to accumulate a specific amount in the future?

◦ It is easier to solve for “n” using the financial calculator or Excel rather than mathematical formulacalculator or Excel rather than mathematical formula.

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Example 5.6 How many years will it take for an investment of $7,500 to grow to $23,000 if it is invested at 8% annually?

Using a Financial CalculatorEnter:

I/Y = 8 PMT = 0PV 7 500PV = -7,500FV = 23,000N = 14.56

Solving for the Number of Periods, nLet’s assume that the Toyota Corporation has guaranteed that the price of a new Prius will always be $20,000, and you’d like to buy one but currently have only $7,752. How many years will it take for your initial investment of $7,752 to grow to $20,000 if it is invested so that it earns 9% compounded annually?if it is invested so that it earns 9% compounded annually?

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How many years will it take for $10,000 to grow to $200,000 given a 15% compound growth rate?

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i=15%Years

Cash flow -$10,000 $200,000

0 1 2 … N =?

We know FV,PV, and i and are solving for

N

In this problem, we are solving for “n”. We know the interest rate, the present value and the future value.

l l “ ” f l We can calculate “n” using a financial calculator or an Excel spreadsheet.

Using a Financial CalculatorEnter:

I/Y = 15 PMT = 0PV 10 000PV = -10,000FV = 200,000N = 21.43

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It will take 21.43 years for $10,000 to grow to $200,000 at an annual interest rate of 15%.

Rule of 72 is an approximate formula to determine the number of years it will take to double the value of your investment.

l f Rule of 72N = 72/interest rate

Key Question: What rate of interest will allow your investment to grow to a desired future value?

d h f We can determine the rate of interest using mathematical equation, the financial calculator or the Excel spread sheet.

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Example 5.8 At what rate of interest must your savings of $10,000 be compounded annually for it to grow to $22,000 in 8 years?

Using Mathematical Formula:I = (FV/PV)1/n - 1

= (22000/10000)1/8 - 1= (2.2).125 - 1= 1 1035 – 1= 1.1035 – 1= .1035 or 10.35%

Using Financial CalculatorEnter:

N = 8 PMT = 0PV 10 000PV = -10,000FV = 22,000I/Y =10.36

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Solving for the Interest Rate, iLet’s go back to that Prius example in Checkpoint 5.5. Recall that the Prius always costs $20,000. In 10 years, you’d really like to have $20,000 to buy a new Prius, but you only have $11,167 now. At what rate must your $11,167 be compounded annually for it to grow to $20 000 in 10 years?for it to grow to $20,000 in 10 years?

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At what rate will $50,000 have to grow to reach $1,000,000 in 30 years?

i=?%

Years

Cash flow -$50,000 $1,000,000

0 1 2 … 30

We know FV, PV and N and areSolving for “interestrate”

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Here we are solving for the interest rate. The number of years, the present value, the future value are known.

h We can compute the interest rate using mathematical formula, a financial calculator or an Excel spreadsheet.

Using Mathematical FormulaI = (FV/PV)1/n - 1

= (1000000/50000)1/30 - 1= (20)0.0333 - 1= 1 1050 -1= 1.1050 -1= .1050 or 10.50%

You will have to earn an annual interest rate of 10.5% for 30 years to increase the value of investment from $50,000 to $1,000,000.

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The annual percentage rate (APR) indicates the amount of interest paid or earned in one year without compounding.

APR is also known as the nominal or stated i t t tinterest rate.

This is the rate required by law.

We cannot compare two loans based on APR if they do not have the same compounding period.

To make them comparable, we calculate their equivalent rate using an annual compounding period We do this by calculating the effectiveperiod. We do this by calculating the effective annual rate (EAR)

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Example 5.9 Calculate the EAR for a loan that has a 5.45% quoted annual interest rate compounded monthly.

[ ]12 EAR = [1+.0545/12]12 - 1= 1.0558 – 1= .05588 or 5.59%

Calculating an EAR or Effective Annual RateAssume that you just received your first credit card statement and the APR, or annual percentage rate listed on the statement, is 21.7%. When you look closer you notice that the interest is compounded daily. What is the EAR, or effective annual rate, on your credit card?

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What is the EAR on a quoted or stated rate of 13% that is compounded monthly?

i= an annual rate of 13% that is compounded monthly Months

0 1 2 … 12

Compounding periodsare expressed in months (i.e. m=12) and we are Solving for EAR

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Here we need to solve for Effective Annual Rate (EAR).

We can compute the EAR using equation 5.4

EAR = [1+.13/12]12 - 1= 1.1380 – 1

= .1380 or 13.80%

There is a significant difference between APR and EAR.

If the interest rate is not compounded ll h ld hannually, we should compute the EAR to

determine the actual interest earned on an investment or the actual interest paid on a loan.

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When the time intervals between when interest is paid are infinitely small, we can use the following mathematical formula to compute the EAR.

EAR = (e quoted rate ) – 1◦ Where “e” is the number 2.71828

Example 5.10 What is the EAR on your credit card with continuous compounding if the APR is 18%?

18 EAR = e.18 - 1 = 1.1972 – 1 = .1972 or 19.72%